揭示高容量核霍普菲尔德网络中吸引子景观的自组织机制。
Self-Organization and Spectral Mechanism of Attractor Landscapes in High-Capacity Kernel Hopfield Networks
- 提出峰值锐度度量，发现优化脊线上的吸引子稳定性
- 在高负载下实现最大鲁棒性，支持高记忆容量
- 适合研究神经网络动力学与记忆机制的学者
基于核的学习方法可显著提升霍普菲尔德网络的存储容量，但其背后的动态机制仍不清晰。本文结合吸引子景观的几何表征与核机器的谱理论，提出新度量Pinnacle Sharpness，实证揭示了吸引子稳定性的丰富相图，识别出在高负载条件下实现最大鲁棒性的优化脊线。该脊线表现为力拮抗现象：强驱动力被集体反馈力抵消。理论上，此行为源于权重谱的特定重组，称为谱集中。不同于简单的秩1坍缩，网络在脊线上自组织为临界态：主特征值被放大以增强全局稳定性（直接力），尾部特征值保持有限以维持高记忆容量（间接力）。这些结果表明，学习通过谱机制协调了高维关联记忆模型中的稳定性和容量。
原文摘要 · Abstract (English)
Kernel-based learning methods can dramatically increase the storage capacity of Hopfield networks, yet the dynamical mechanisms behind this enhancement remain poorly understood. We address this gap by combining a geometric characterization of the attractor landscape with the spectral theory of kernel machines. Using a novel metric, Pinnacle Sharpness, we empirically uncover a rich phase diagram of attractor stability, identifying a Ridge of Optimization where the network achieves maximal robustness under high-load conditions. Phenomenologically, this ridge is characterized by a Force Antagonism, in which a strong driving force is counterbalanced by a collective feedback force. We theoretically interpret this behavior as a consequence of a specific reorganization of the weight spectrum, which we term Spectral Concentration. Unlike a simple rank-1 collapse, our analysis shows that the network on the ridge self-organizes into a critical regime: the leading eigenvalue is amplified to enhance global stability (Direct Force), while the trailing eigenvalues remain finite to sustain high memory capacity (Indirect Force). Together, these results suggest a spectral mechanism by which learning reconciles stability and capacity in high-dimensional associative memory models.
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